The coordinate ring of the k-fold iterated commutator locus for 2x2 matrices
Abstract
For 2 × 2 matrices A1,…,Ak, write [A1,…,Ak] for the left-normed iterated commutator […[[A1,A2],A3],…,Ak], and Ik for the ideal, in the 3k-variable reduced-coordinate polynomial ring Rk, cutting out its vanishing locus. We prove, for every k ≥ 2 over any field of characteristic ≠ 2, and as four independently-established results rather than one bundled claim: Ik has codimension 2; Ik has exactly 3 minimal generators; Rk / Ik is Cohen-Macaulay; and Ik is radical. The last of these, together with an explicit component count resting on a non-containment argument, assembles into the Primary Decomposition Theorem: Ik = P2 ∩ ⋯ ∩ Pk is an irredundant primary decomposition into exactly k−1 primes, following an explicit recursive block-involvement pattern. The proof identifies Ik as the ideal of 2 × 2 minors of an explicit 2 × 3 matrix (a determinantal ideal, not merely one that looks determinantal), and invokes classical determinantal-ideal theory (Bruns-Vetter) and an explicit rank-2 Jacobian witness on every component (Serre's criterion) for the algebraic and radicality halves respectively.
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BibTeX
@misc{snellman2026coordinate,
title = {{The coordinate ring of the k-fold iterated commutator locus for 2x2 matrices}},
author = {Jan Snellman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05386},
eprint = {2609.05386},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05386},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The coordinate ring of the k-fold iterated commutator locus for 2x2 matrices
AU - Jan Snellman
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05386
UR - https://arxiv.org/abs/2609.05386
AB - For 2 × 2 matrices A_1,…,A_k, write [A_1,…,A_k] for the left-normed iterated commutator […[[A_1,A_2],A_3],…,A_k], and I_k for the ideal, in the 3k-variable reduced-coordinate polynomial ring R_k, cutting out its vanishing locus. We prove, for every k ≥ 2 over any field of characteristic ≠ 2, and as four independently-established results rather than one bundled claim: I_k has codimension 2; I_k has exactly 3 minimal generators; R_k / I_k is Cohen-Macaulay; and I_k is radical. The last of these, together with an explicit component count resting on a non-containment argument, assembles into the Primary Decomposition Theorem: I_k = P_2 ∩ ⋯ ∩ P_k is an irredundant primary decomposition into exactly k−1 primes, following an explicit recursive block-involvement pattern. The proof identifies I_k as the ideal of 2 × 2 minors of an explicit 2 × 3 matrix (a determinantal ideal, not merely one that looks determinantal), and invokes classical determinantal-ideal theory (Bruns-Vetter) and an explicit rank-2 Jacobian witness on every component (Serre's criterion) for the algebraic and radicality halves respectively.
ER -